On entropic uncertainty relations for measurements of energy and its "complement". (arXiv:1807.11413v3 [quant-ph] UPDATED)

Heisenberg's uncertainty principle in application to energy and time is a
powerful heuristics. This statement plays the important role in foundations of
quantum theory and statistical physics. If some state exists for a finite
interval of time, then it cannot have a completely definite value of energy. It
is well known that the case of energy and time principally differs from more
familiar examples of two non-commuting observables. Since quantum theory was
originating, many approaches to energy-time uncertainties have been proposed.
Entropic way to formulate the uncertainty principle is currently the subject of
active researches. Using the Pegg concept of complementarity of the
Hamiltonian, we obtain uncertainty relations of the "energy-time" type in terms
of the R\'{e}nyi and Tsallis entropies. Although this concept is somehow
restricted in scope, derived relations can be applied to systems typically used
in quantum information processing. Both the state-dependent and
state-independent formulations are of interest. Some of the derived
state-independent bounds are similar to the results obtained within a more
general approach on the basis of sandwiched relative entropies. The developed
method allows us to address the case of detection inefficiencies.

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